We used rest-frame \(EW(\mbox{Ly$\alpha $})\)  measurements to confirm the presence of LAHs around non-LAEs in Section REF  and to compare the \(X_{\rm LAE}\)  with a given \(EW(\mbox{Ly$\alpha $})\)  cut for that in [1]} in Section REF  in order to validate that our sample is not biased s the LAE selection. Therefore, the \(EW(\mbox{Ly$\alpha $})\)  values in this paper were measured in the galaxy's stellar-component scale following [1]}. Here we give a brief summary of the methods for the \(EW(\mbox{Ly$\alpha $})\)  measurements and the \(X_{\rm LAE}\)  calculations. We note that investigating \(\mbox{Ly$\alpha $}\)  line properties including \(EW(\mbox{Ly$\alpha $})\)  for our individual sources is beyond the scope of this paper. We plan to improve the method for measuring \(EW(\mbox{Ly$\alpha $})\)  in the MUSE 3D cube and discuss them in a future project.
